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Passage 2
P
W centre of mass about an axis, and (ii) its motion about an instantaneous axis passing through the centre of mass. These axes need not be stationary. Consider, for example, a thin uniform disc welded (rigidly fixed) horizontally at its rim to a massless stick, as shown in the figure. When the disc-stick system is rotated about the origin on a horizontal frictionless plane with angular speed \( \omega \), the motion at any instant can be taken as a combination of (i) a rotation of the centre of mass of the disc about the \( Z \)-axis, and (ii) a rotation of the disc through an instantaneous vertical axis passing through its centre of mass (as is seen from the changed orientation of points \( P \) and Q). Both these motions have the same angular speed \( \omega \) in this case.
Now, consider two similar systems as shown in the figure : Case (a) the disc with its face vertical and parallel to \( x-z \) plane; Case (b) the disc with its face making an angle of \( 45^{\circ} \) with \( x \) - \( y \) plane and its horizontal diameter parallel to \( X \)-axis. In both the cases, the disc is welded at point \( P \), and the systems are rotated with constant angular speed \( \omega \) about the \( Z \)-axis.

Which of the following statements regarding the angular speed about the instantaneous axis (passing through the centre of mass) is correct?
(2012)
(a) It is \( \sqrt{2} \omega \) for both the cases
(b) It is \( \omega \) for case (a); and \( \frac{\omega}{\sqrt{2}} \) for case (b)
(c) It is \( \omega \) for case (a); and \( \sqrt{2} \omega \) for case (b)
(d) It is \( \omega \) for both the cases
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